Rings and Resonances

Ring Systems

All four of the large outer planets have rings. Ring systems obey very strict
geometrical laws:

Orbits of ring particles must be almost perfectly circular. Otherwise collisions will
rapidly destroy the ring.

Ring particle orbits must lie precisely in the equatorial plane of the planet. Inclined
orbits precess, or change their orientation. Very quickly, an inclined ring will
soon degenerate into swarms of particles with different orbital planes.

The requirement to orbit in the equatorial plane of the planet means that rings are
always extremely flat.

Because all the ringed planets have numerous large satellites, the orbits of ring
particles are constantly being disturbed. Thus collisions must still occur. Implications:

Rings are dynamic and probably short-lived in geologic terms. At other times in Solar
System history, perhaps Jupiter or Neptune had the grand ring system and Saturn's was
skimpy or absent.

Some mechanism must be maintaining the sharp edges of observed rings. The most likely
mechanism is shepherd moons (see below).

Resonances result when two celestial objects interact with each other gravitationally
at regular intervals. The regularity of the interaction can do one of two things:

Lock the two objects in step so they repeat the same patterns of movement.

Perturb one or both bodies enough to break up the resonance.

We find examples of both in the Solar System.

Tidal Rotation Lock

A tidal rotation lock is the simplest kind of resonance. Planets and their satellites
cause tidal bulges in one another. As the objects rotate, the tidal bulges move through
their interiors and encounter friction. The satellites, being smaller, tend to become
locked first. Some results:

Almost all satellites in the Solar System show the same face to their planet, as the
Moon does to Earth.

Satellites also slow their planets' rotation.

The satellites of the Jovian planets are so small in comparison to their planets that
their effect is negligible. The moons of Mars are so tiny they have negligible effect
also.

The Earth, with a satellite 1% as massive, has been slowed noticeably: from 18 hours
rotation 900 million years ago to 24 hours today.

Pluto, with a satellite 10-15% as massive and much closer than our Moon, has become
completely locked. Pluto and Charon always show the same face to each other.

Mercury

For a long time, astronomers thought Mercury was tidally locked to the Sun as the Moon
is to Earth. It actually has a more complex tidal rotation lock; Mercury rotates in 56
days; three times for every two orbits. Some consequences:

There are two "hot poles" on Mercury where the Sun is directly overhead when
the planet is at perihelion. One is the Caloris Basin, the great impact crater (hence the
name); the other is directly opposite.

At perihelion, Mercury does very nearly rotate at the same speed as it goes around the
Sun (probably explaining the 3/2 resonance - tidal forces on Mercury are about 5 times
greater at perihelion). You would see the Sun stand still, reverse slightly, then go
forward again. If you were on the day-night boundary, you would see the Sun rise, set
briefly in the east, then rise again.

Mercury's equator points exactly at the Sun because of the intense tidal forces. Thus
the poles never get more than grazing illumination and are quite cold. Polar craters never
get sun at all and actually seem to contain ice.

Long-Term Effects

As a planet rotates, friction causes its tidal bulges to lead its satellite. The
satellite pulls on the near tidal bulge, slowing the planet. The near tidal bulge pulls on
the satellite, accelerating it and throwing it into a more distant orbit.

For the Earth-Moon system, the two effects are predicted to balance several billion
years in the future, when the Moon will be about 50% farther away and the day and month
will both be 40 of our present days long. But that's not the end of the story. The Sun
also exerts tidal forces on the Earth, about half as strong as the Moon. The Sun will
continue to slow Earth's rotation; the solar tidal bulges will trail the moon,
slowing it down and causing it to spiral inward. Whether the Moon will spiral in enough to
break up from tidal forces before the Sun goes red giant is unknown.

If the satellite orbits opposite its planet's rotation, the tidal bulges always trail
the satellite. They tend to drag the satellite in and may eventually cause it to be
destroyed. Neptune's large moon Triton is the only body in the Solar System in this
category; it may spiral into Neptune in a few hundred million to several billion years.

The Roche Limit and Tidal Disruption

Tidal forces result when a celestial body exerts a stronger gravitational pull on the
near side of another body than the far side. The difference can be great enough to pull
fragile objects apart, as Jupiter did to Comet Schumacher-Levy in 1992. Within a certain
distance of any planet, called the Roche Limit, tidal forces are greater than the
gravitational attraction of small bodies and satellites cannot form.

It is possible for satellites to orbit within the Roche Limit (Our own artificial
satellites do; Mars and the Jovian planets have satellites that do) but the satellite is
held together by the strength of its materials and not by its gravity.

If a natural satellite is orbiting within its planet's Roche limit, it must have formed
elsewhere and perturbed into a close orbit somehow.

There has been speculation that planetary rings may have formed from tidally-disrupted
satellites. Possibly Triton and our Moon will be disrupted if they eventually come within
the Roche Limit.

Orbital Resonances

Gaps

A asteroid orbiting 3.28 AU from the Sun would circle the Sun in 5.93 years, half the
orbital period of Jupiter. Every other revolution, it would receive a tug from Jupiter at
the same point in its orbit and eventually have its orbit changed. There are thus no
asteroids with periods exactly half that of Jupiter, and almost none with a period 1/3
that of Jupiter. These gaps in the asteroid belt are called Kirkwood Gaps.
Principal Kirkwood gaps correspond to orbital periods 1/3, 2/5, 3/7, 1/2 and 3/5 that of
Jupiter.

Gaps occur in Saturn's rings due to Saturn's satellites. Particles moving within
Cassini's Division would orbit Saturn in periods ranging from 11 hr. 19 min. to 12 hr. 5
min. This is 1/2 the period of the satellite Mimas, 1/3 that of Enceladus, 1/4 that of
Tethys and 1/9 that of Rhea. Smaller gaps have been noted in the rings and are due to
other resonances.

Shepherd Moons

Consider a moon (it can be tiny) orbiting just outside a ring. Ring particles, being
closer to the planet than the satellite, are moving faster. If a particle drifts out of
the ring, when it overtakes the satellite, the satellite's gravity will drag the particle
back, slow it down, and cause it to fall back into the ring.

Similarly, a moon can orbit just inside a ring. The satellite is moving faster than the
ring particles. If a particle drifts inward from the ring, it will be overtaken by the
satellite. The satellite's gravity will speed the particle up and throw it back into the
ring.

Saturn has a thin ring (the F ring) bounded by two tiny moons that keep it confined.
Some of the shepherd moons for the thin rings of Uranus and Neptune are also known.

Preferred Periods

Certain resonances seem to enhance orbital stability by locking bodies in step in such
a way they avoid conflict. 3:2 resonances seem to be especially effective. Pluto crosses
Neptune's orbit, but its period is 3/2 that of Neptune, so the two objects never approach
closely. Of the 40-plus objects discovered orbiting beyond Neptune since 1992, an
astounding 40% have periods very close to Pluto's and are also in 3:2 resonance with
Neptune. These objects have been dubbed "plutinos".

In contrast to the gaps in the asteroid belt at 1/2 and 1/3 the period of Jupiter,
there is a cluster of asteroids orbiting with 3/2 Jupiter's period and a smaller cluster
at 4/3.

Lagrangian Points

Johannes Kepler and Isaac Newton solved the equations for orbital motion of one body
around another. For centuries after, astronomers struggled to solve the "three-body
problem": write an equation for the motion of three orbiting bodies. We now believe
such a solution is impossible; in fact, most such systems are chaotic. That is,
tiny differences in initial calculations lead to enormous differences later. The best we
can do in such a system is what we do with the planets now: achieve reasonable precision
over a reasonable time span.

In the late 1700's, the French mathematician Joseph Lagrange solved the three-body
problem for certain special cases. In a system consisting of a Primary body (greatest
mass), a Secondary (intermediate mass) and a third small object, he showed that there were
five points where the third body's motion was predictable.

Points L1 and L2 correspond to orbits around the secondary that match the secondary's
period about the primary, for example, a satellite orbiting Earth about 5 times farther
than the Moon would have a period of a year. If it orbited in the same direction as Earth
in the L2 position or opposite in the L1, it could remain in line with the Earth and Sun.
Position L3 orbits opposite the secondary but a bit closer because it feels the combined
pull of both the Primary and Secondary.

Positions L1, L2 and L3 apply to an ideal system of three bodies and perfectly circular
orbits. In the real Solar System they are unstable: any disturbance would cause the third
body to drift out of position. Solar observatories have been placed at the L1
position of Earth, but they need a fuel supply to correct for the tendency to drift.
In 1985 NASA took advantage of that on-board fuel to divert a probe at
the L1 position into a flyby of Comets Giacobini-Zinner and Halley.

Positions L4 and L5 are different. These points form equilateral triangles with the
Primary and Secondary. L4 leads the Secondary and L5 trails it. These points are
stable. If an object in one of these points is disturbed, it tends to return to its
original position. In real life bodies oscillate fairly widely around the stable point

These points were mathematical abstractions until real examples were found. Asteroids
occupy the L4 and L5 positions on Jupiter's orbit. The first one discovered was 588
Achilles, so the custom arose of naming these asteroids after figures from the Trojan War.
For this reason, objects travelling in an L4 or L5 position are often called Trojan.
In the Saturn system, Tethys and Dione have small moons in their L4 or L5 positions. There
appear to be faint clouds of dust particles in the L4 and L5 positions of our Moon's
orbit. So far no asteroids have been found in the L4 or L5 positions in either Earth's or
Saturn's orbit, despite careful searches.

Chaos

Chaos is the opposite of resonance in a way, and subject to a lot of misconceptions.

Chaos does not mean:

Chaos does mean:

Events are not governed by laws of nature.

Events are governed by the laws of nature but their behavior is too complex for simple
description.

Events are random or unpredictable.

Events may be predictable in the short term but not over arbitrarily long times. Small
errors at the start of a prediction compound later on into great differences.

Chaotic systems are unstable and will eventually fall apart (the Jurassic Park
Fallacy)

Chaotic systems may remain within fixed bounds but their behavior within those bounds
may be hard to predict.

Saturn's moon Hyperion has a thick disk shape that has been likened to a hockey puck.
Because of its asymmetry, gravitational interactions with other Saturnian satellites cause
it to rotate chaotically. That is, there is no single rotation axis and period that
describes Hyperion's rotation for long periods of time.

Three- and larger body systems (and hence the Solar System) are probably chaotic. That
is, we could not predict exactly.where the planets will be in exactly a billion years, or
say where they were exactly a billion years ago. The only way to predict motions of the
planets with extreme accuracy now is to calculate all their gravitational interactions and
motions in small steps; tiny errors in our knowledge of masses and distances now would
compound into huge errors in billions of years. We can accurately reconstruct celestial
events in ancient documents, but on a scale of millions or billions of years our
predictions would be increasingly inaccurate. Note that this does not necessarily
mean the Solar System is unstable!